I cannot for the life of me do this even with the hint and I just need some clear, easy to read step by step on how to do it. Please write on paper as the symbols on here can get very messy. Thankyou very much

= Let X1, X2, ... be i.i.d. copies of a positive random variable X satisfying P(X > 0) = 1 and E(X2) 0 and variance o2 = Var(X) > 0. Define () n Sn = xi, = n=1,2,..., i=1 with So = 0. For k > 0, define Tk = inf {n > 0: Sn> k}. Show that Tk Ta k/a dy z as k + 0, where 2 ~ N(0,0?/a). Z Tk-k/ Hints: Express P(Tula 5 z) as P(Samaran 2 y) for some n and some y depending on k -k/ and z, and show that P(Txala 5 z) + P(Q Jk) + P(Q> 9) where Q ~ N(0,0%). The normal random variable Q ~ N(0,02) is symmetric around 0, with P(Q 2 y) = PQ 0) = 1 and E(X2) 0 and variance o2 = Var(X) > 0. Define () n Sn = xi, = n=1,2,..., i=1 with So = 0. For k > 0, define Tk = inf {n > 0: Sn> k}. Show that Tk Ta k/a dy z as k + 0, where 2 ~ N(0,0?/a). Z Tk-k/ Hints: Express P(Tula 5 z) as P(Samaran 2 y) for some n and some y depending on k -k/ and z, and show that P(Txala 5 z) + P(Q Jk) + P(Q> 9) where Q ~ N(0,0%). The normal random variable Q ~ N(0,02) is symmetric around 0, with P(Q 2 y) = PQ